Let N g (d) be the set of primes p such that the order of g modulo p is divisible by a prescribed... more Let N g (d) be the set of primes p such that the order of g modulo p is divisible by a prescribed integer d. K. Wiertelak [Acta Arith. 43, 177–190 (1984; Zbl 0531.10049)] showed that this set has a natural density and gave a rather involved explicit expression for it. Let N g (d)(x) be the number of primes p≤x that are in N g (d). A simple identity for N g (d)(x) is established. It is used to derive a more compact expression for the natural density than known hitherto. A numerical demonstration, using a program of Y. Gallot, is presented.
An infinite sequence $\langle{u_n}\rangle_{n\in\mathbb{N}}$ of real numbers is holonomic (also kn... more An infinite sequence $\langle{u_n}\rangle_{n\in\mathbb{N}}$ of real numbers is holonomic (also known as P-recursive or P-finite) if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is said to be positive if each $u_n \geq 0$, and minimal if, given any other linearly independent sequence $\langle{v_n}\rangle_{n \in\mathbb{N}}$ satisfying the same recurrence relation, the ratio $u_n/v_n$ converges to $0$. In this paper, we focus on holonomic sequences satisfying a second-order recurrence $g_3(n)u_n = g_2(n)u_{n-1} + g_1(n)u_{n-2}$, where each coefficient $g_3, g_2,g_1 \in \mathbb{Q}[n]$ is a polynomial of degree at most $1$. We establish two main results. First, we show that deciding positivity for such sequences reduces to deciding minimality. And second, we prove that deciding minimality is equivalent to determining whether certain numerical expressions (known as periods, exponential periods, and period-like integrals) are equal to zero. Period...
Let $\zeta_q$ be a primitive $q^{\text{th}}$ root of unity with $q$ an arbitrary odd prime. The r... more Let $\zeta_q$ be a primitive $q^{\text{th}}$ root of unity with $q$ an arbitrary odd prime. The ratio of Kummer's first factor of the class number of the cyclotomic number field $\mathbb{Q}(\zeta_q)$ and its expected order of magnitude (a simple function of $q$) is called the Kummer ratio and denoted by $r(q)$. It is known that typically $r(q)$ is close to 1, but nevertheless it is believed that it is unbounded, but only large on a very thin sequence of primes $q$. We propose an algorithm to compute $r(q)$ requiring the evaluation of $O(q\log q)$ products and $O(q)$ logarithms. Using it we obtain a new record maximum for $r(q)$, namely $r(6766811) =1.709379\dotsc$ (the old record being $r(5231)=1.556562\dotsc$). The program used and the results described here, are collected at the following address \url{this http URL}. This is a (preliminary) report about the computational part of a joint project with Pieter Moree, Sumaia Saad Eddin, and Alisa Sedunova.
Let a_n(k) be the kth coefficient of the nth cyclotomic polynomial Phi_n(x). As n ranges over the... more Let a_n(k) be the kth coefficient of the nth cyclotomic polynomial Phi_n(x). As n ranges over the integers, a_n(k) assumes only finitely many values. For any such value v we determine the density of integers n such that a_n(k)=v. Also we study the average of the a_n(k). We derive analogous results for the kth Taylor coefficient of 1/Phi_n(x) (taken around x=0), the kth coefficient of the nth reciprocal cyclotomic polynomial. We formulate various open problems.
The convolved Fibonacci numbers F_j^(r) are defined by (1-z-z^2)^{-r}=\sum_{j>=0}F_{j+1}^(r)z^... more The convolved Fibonacci numbers F_j^(r) are defined by (1-z-z^2)^{-r}=\sum_{j>=0}F_{j+1}^(r)z^j. In this note some related numbers that can be expressed in terms of convolved Fibonacci numbers are considered. These numbers appear in the numerical evaluation of a certain number theoretical constant. This note is a case study of the transform {1/n}\sum_{d|n}mu(d)f(z^d)^{n/d}, with f any formal series and mu the Moebius function), which is studied in a companion paper entitled `The formal series Witt transform'.
Bulletin of the Australian Mathematical Society, 2019
Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb{Z}/p\mathbb{... more Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb{Z}/p\mathbb{Z})^{\ast },$ then we say that $g$ is a $t$ -near primitive root modulo $p$ . We point out the easy result that each coprime residue class contains a subset of primes $p$ of positive natural density which do not have $g$ as a $t$ -near primitive root and we prove a more difficult variant.
A small survey of work done on estimating the number of integers without large prime factors up t... more A small survey of work done on estimating the number of integers without large prime factors up to around 1950 is provided. Around 1950 N.G. de Bruijn published results that dramatically advanced the subject and started a new era in this topic.
Let a_1,...,a_m be positive real numbers. Besser and Moree considered weighted numbers of -1,+1 s... more Let a_1,...,a_m be positive real numbers. Besser and Moree considered weighted numbers of -1,+1 solutions of the linear inequality |a_i-a_j| < e_ka_k < a_i+a_j, with e_k=-1 of 1 and k running over the integers 1,...,m with i and j skipped. They introduced some invariants and near invariants related to this situation (invariant meaning here: not depending on the choice of i and j). The main result of their paper is extended here to a much more general setting, namely that of certain maps from finite sets to {-1,1}. Some applications are given.
Based on congruences mod p and on properties of Bernoulli polynomials and numbers, several condit... more Based on congruences mod p and on properties of Bernoulli polynomials and numbers, several conditions are derived for x; k 2 if they satisfy the diophantine equation 1 k + 2 k + \Delta \Delta \Delta + (x \Gamma 1) k = x k . By using the results of experiments with these conditions on an SGI workstation it is proved that x can not be divisible by any irregular prime ! 10000 and that k is divisible by the least common multiple of all the integers 200. Moreover, it is proved that x can not be divisible by any regular prime.
Among other things we show that for each n-tuple of positive rational numbers (a 1 ; : : : ; a n ... more Among other things we show that for each n-tuple of positive rational numbers (a 1 ; : : : ; a n ) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a 1 x 1 + +a n x n = 1 with x 1 ; : : : ; x n S-units are not contained in fewer than exp((4 + o(1))s 1=2 (log s) 1=2 ) proper linear subspaces of C n . This generalizes a result of Erdős, Stewart and Tijdeman [7] for S-unit equations in two variables. Further, we prove that for any algebraic number field K of degree n, any integer m with 1 m < n, and any sufficiently large s there are integers 0 ; : : : ; m in K which are linearly independent over Q , and prime numbers p 1 ; : : : ; p s , such that the norm polynomial equation jN K=Q ( 0 + 1 x 1 + + mxm )j = p z1 1 p zs s has at least expf(1+o(1)) n m s m=n (log s) 1+m=n g solutions in x 1 ; : : : ; xm ; z 1 ; : : : ; z s 2 Z. This generalizes a result of Moree and Stewart [19] for m = 1. Our main tool, also established in this pap...
Let N g (d) be the set of primes p such that the order of g modulo p is divisible by a prescribed... more Let N g (d) be the set of primes p such that the order of g modulo p is divisible by a prescribed integer d. K. Wiertelak [Acta Arith. 43, 177–190 (1984; Zbl 0531.10049)] showed that this set has a natural density and gave a rather involved explicit expression for it. Let N g (d)(x) be the number of primes p≤x that are in N g (d). A simple identity for N g (d)(x) is established. It is used to derive a more compact expression for the natural density than known hitherto. A numerical demonstration, using a program of Y. Gallot, is presented.
An infinite sequence $\langle{u_n}\rangle_{n\in\mathbb{N}}$ of real numbers is holonomic (also kn... more An infinite sequence $\langle{u_n}\rangle_{n\in\mathbb{N}}$ of real numbers is holonomic (also known as P-recursive or P-finite) if it satisfies a linear recurrence relation with polynomial coefficients. Such a sequence is said to be positive if each $u_n \geq 0$, and minimal if, given any other linearly independent sequence $\langle{v_n}\rangle_{n \in\mathbb{N}}$ satisfying the same recurrence relation, the ratio $u_n/v_n$ converges to $0$. In this paper, we focus on holonomic sequences satisfying a second-order recurrence $g_3(n)u_n = g_2(n)u_{n-1} + g_1(n)u_{n-2}$, where each coefficient $g_3, g_2,g_1 \in \mathbb{Q}[n]$ is a polynomial of degree at most $1$. We establish two main results. First, we show that deciding positivity for such sequences reduces to deciding minimality. And second, we prove that deciding minimality is equivalent to determining whether certain numerical expressions (known as periods, exponential periods, and period-like integrals) are equal to zero. Period...
Let $\zeta_q$ be a primitive $q^{\text{th}}$ root of unity with $q$ an arbitrary odd prime. The r... more Let $\zeta_q$ be a primitive $q^{\text{th}}$ root of unity with $q$ an arbitrary odd prime. The ratio of Kummer's first factor of the class number of the cyclotomic number field $\mathbb{Q}(\zeta_q)$ and its expected order of magnitude (a simple function of $q$) is called the Kummer ratio and denoted by $r(q)$. It is known that typically $r(q)$ is close to 1, but nevertheless it is believed that it is unbounded, but only large on a very thin sequence of primes $q$. We propose an algorithm to compute $r(q)$ requiring the evaluation of $O(q\log q)$ products and $O(q)$ logarithms. Using it we obtain a new record maximum for $r(q)$, namely $r(6766811) =1.709379\dotsc$ (the old record being $r(5231)=1.556562\dotsc$). The program used and the results described here, are collected at the following address \url{this http URL}. This is a (preliminary) report about the computational part of a joint project with Pieter Moree, Sumaia Saad Eddin, and Alisa Sedunova.
Let a_n(k) be the kth coefficient of the nth cyclotomic polynomial Phi_n(x). As n ranges over the... more Let a_n(k) be the kth coefficient of the nth cyclotomic polynomial Phi_n(x). As n ranges over the integers, a_n(k) assumes only finitely many values. For any such value v we determine the density of integers n such that a_n(k)=v. Also we study the average of the a_n(k). We derive analogous results for the kth Taylor coefficient of 1/Phi_n(x) (taken around x=0), the kth coefficient of the nth reciprocal cyclotomic polynomial. We formulate various open problems.
The convolved Fibonacci numbers F_j^(r) are defined by (1-z-z^2)^{-r}=\sum_{j>=0}F_{j+1}^(r)z^... more The convolved Fibonacci numbers F_j^(r) are defined by (1-z-z^2)^{-r}=\sum_{j>=0}F_{j+1}^(r)z^j. In this note some related numbers that can be expressed in terms of convolved Fibonacci numbers are considered. These numbers appear in the numerical evaluation of a certain number theoretical constant. This note is a case study of the transform {1/n}\sum_{d|n}mu(d)f(z^d)^{n/d}, with f any formal series and mu the Moebius function), which is studied in a companion paper entitled `The formal series Witt transform'.
Bulletin of the Australian Mathematical Society, 2019
Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb{Z}/p\mathbb{... more Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb{Z}/p\mathbb{Z})^{\ast },$ then we say that $g$ is a $t$ -near primitive root modulo $p$ . We point out the easy result that each coprime residue class contains a subset of primes $p$ of positive natural density which do not have $g$ as a $t$ -near primitive root and we prove a more difficult variant.
A small survey of work done on estimating the number of integers without large prime factors up t... more A small survey of work done on estimating the number of integers without large prime factors up to around 1950 is provided. Around 1950 N.G. de Bruijn published results that dramatically advanced the subject and started a new era in this topic.
Let a_1,...,a_m be positive real numbers. Besser and Moree considered weighted numbers of -1,+1 s... more Let a_1,...,a_m be positive real numbers. Besser and Moree considered weighted numbers of -1,+1 solutions of the linear inequality |a_i-a_j| < e_ka_k < a_i+a_j, with e_k=-1 of 1 and k running over the integers 1,...,m with i and j skipped. They introduced some invariants and near invariants related to this situation (invariant meaning here: not depending on the choice of i and j). The main result of their paper is extended here to a much more general setting, namely that of certain maps from finite sets to {-1,1}. Some applications are given.
Based on congruences mod p and on properties of Bernoulli polynomials and numbers, several condit... more Based on congruences mod p and on properties of Bernoulli polynomials and numbers, several conditions are derived for x; k 2 if they satisfy the diophantine equation 1 k + 2 k + \Delta \Delta \Delta + (x \Gamma 1) k = x k . By using the results of experiments with these conditions on an SGI workstation it is proved that x can not be divisible by any irregular prime ! 10000 and that k is divisible by the least common multiple of all the integers 200. Moreover, it is proved that x can not be divisible by any regular prime.
Among other things we show that for each n-tuple of positive rational numbers (a 1 ; : : : ; a n ... more Among other things we show that for each n-tuple of positive rational numbers (a 1 ; : : : ; a n ) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a 1 x 1 + +a n x n = 1 with x 1 ; : : : ; x n S-units are not contained in fewer than exp((4 + o(1))s 1=2 (log s) 1=2 ) proper linear subspaces of C n . This generalizes a result of Erdős, Stewart and Tijdeman [7] for S-unit equations in two variables. Further, we prove that for any algebraic number field K of degree n, any integer m with 1 m < n, and any sufficiently large s there are integers 0 ; : : : ; m in K which are linearly independent over Q , and prime numbers p 1 ; : : : ; p s , such that the norm polynomial equation jN K=Q ( 0 + 1 x 1 + + mxm )j = p z1 1 p zs s has at least expf(1+o(1)) n m s m=n (log s) 1+m=n g solutions in x 1 ; : : : ; xm ; z 1 ; : : : ; z s 2 Z. This generalizes a result of Moree and Stewart [19] for m = 1. Our main tool, also established in this pap...
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